Learning in Recurrent Networks Psychology 209 February 25 & 27, 2013 Outline Back Propagation through time Alternatives that can teach networks to settle to fixed points Learning conditional distributions An application Collaboration of hippocampus & cortex in learning new associations Back Propagation
Through Time Error at each unit is the injected error (arrows) and the back-propagated error; these are summed and scaled by deriv. of activation function to calculate deltas. Continuous back prop through time as implemented in rbp Time is viewed as consisting of intervals from 0 to nintervals (tmax). Inputs clamped typically from t=0 for 2-3 intervals.
Activation equation (for t = t:t:tmax): neti(t)= t ( Sjaj(t-t)wij + bi ) + (1 t) neti(t-t) Calculation of deltas (for t = tmax:-t:t): dj(t) = t ( f(netj(t)) E/aj(t) ) + (1 t) dj(t+t) Where dj(tmax+t) = 0 for all j and E/aj(t) = Skwkjdk(t+t) + (t(t) a(t))
Targets are usually provided over the last 2-3 intervals. Then change weights using: E/wij = St=1:t:tmaxaj(t-1)di(t) Include momentum and weight decay if desired. Use CE instead of E if desired: CE = -Si [tilog(ai) + (1-ti)log(1-ai)]
Recurrent Network Used in Rogers et al Semantic Network Simulation Plusses and Minuses of BPTT Can learn arbitrary trajectories through state space (figure eights, etc). Works very reliably in training networks to settle to desired target states. Biologically implausiblemax Gradient gets very thin over many time steps Several Variants and Alternative Algorithms (all relevant to networks that settle to a fixed point) Almeda/Pineda algorithm Discussed in Williams and Zipser reading along with
many other variants of back prop through time Recirculation and Generec. Discussed in OReilly Reading Contrastive Hebbian Learning. Discussed in Movellan and McClelland reading Almeda Pineda Algorithm (Notation from OReilly, 1996) Update net inputs (h) until they stop changing according to (s(.)(.) = logistic fcn): ji Then update deltas (y) til they stop changing according to: J represents the external error to the
unit, if any. Adjust weights using the delta rule Assuming symmetric connections: jk Only activation is propagated. Time difference of activation reflects error signal. Maybe this is more biologically plausible that explicit backprop of error? Generalized Recirculation
OReilly, 1996 Minus phase: Present input, feed activation forward,compute output, let it feed back, let network settle. tk Plus phase: Then clamp both input and output units into desired state, and let network settle again.* hj, yj si *equations neglect the component
to the net input at the hidden layer from the input layer. A problem for backprop and approximations to it: Average of Two Solutions May not be a Solution Network Must Be Stochastic Boltzmann Machine P(a = 1) = logistic(net/T) Continuous Diffusion Network (g = 1/T), Zi(t) is a sample of Gaussian noise Contrastive Hebbian Learning Rule
Present Input only (minus phase) Settle to equilibrium (change still occurs but distribution stops changing) Do this several times to sample distribution of states at equilibrium Collect coproducts ai-aj-; avg = Present input and targets (plus phase) Collect coproducts ai+aj+; avg = Change weights according to: Dwij = (- )
The contrastive Hebbian learning rule minimizes: The sum, over different input patterns I: of the contrastive divergence or Information Gain between probability distributions over states s of the output units for desired (plus) and obtained (minus) phases, conditional on the Input I p (s | I ) ds TIG p ( s | I ) ln I s p (s | I ) In a continuous diffusion network, probability flows over time until it reaches an equilibrium distribution
Patterns and Distributions Desired Distrib Obtained Results Problems and Solutions Stochastic neural networks are VERY slow to train because you need to settle (which takes many time steps) many times in each of the plus and minus phases to collect adequate statistics. Perhaps RBMs and Deep Networks can help here? Collaboration of Hippocampus and
Neocortex The effects of prior association strength on memory in both normal and control subjects are consistent with the idea that hippocampus and neocortex work synergistically rather than simply providing two different sources of correct performance. Even a damaged hippocampus can be helpful when the prior association is very strong. Performance of Control and Amnesic Patients in Learning Word Pairs with Prior Associations Cutting (1978), Expt. 1 100
Control (Expt) Percent Correct 80 Amnesic (Expt) 60 40 20 0 Base rates -20
Very Easy Easy Fairly Easy Hard Category (Ease of Association) man:woman hungry:thin city:ostrich Very Hard
Kwok & McClelland Model Model includes slow learning cortical system representing the content of an association and the context. Hidden units in neo-cortex mediate associative learning. Cortical network is pre-trained with
several cue-relation-response triples for each of 20 different cues. When tested just with cue as probe it tends to produce different targets with different probabilities: Dog (chews) bone (~.30) Dog (chases) cat (~.05) Then the network is shown cue-responsecontext triples. Hippo. learns fast and cortex learns (very) slowly.
Hippocampal and cortical networks work together at recall, so that even weak hippocampal learning can increase probability of settling to a very strong preexisting association. Hippocampus Context Neo-Cortex Relation Cue
Response Data with Simulation Results From K&M Model Cutting (1978), Expt. 1 100 Control (Model) 84 80 Amnesic (Model) Percent Correct
70 68 Control (Expt) 60 Amnesic (Expt) 40 20 9 0 0
0 -20 Very Easy Easy Fairly Easy Hard Category (Ease of Association) Very Hard
